Check with the local clubs and ranges near you. Sometimes
the do "League" matches, both pistol and rifle.

Around here (Pacific Northwest) there are monthly steel matches
at several ranges, and they have divisions for both rimfire pistol
and rimfire rifle.

Ruger Rimfire matches are popular in some areas.

22 bullseye competitions, 22 silhouette, 22 benchrest, there
are LOTS of games played with the rimfire gun! Get out there
and ask. Local gunshops, indoor and outdoor ranges, someone
will know whats going on locally.